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Файл:The design of the exoskeleton. Monograph
.pdf
81
The simulation results showed that the oscillations of the motor
torque due to the dependence of the rotor and stator mutual inductance
on the angular position of the rotor in the conventional control method can
be up to 20% of the "working" torque. For exoskeletons used in medical
practice for the rehabilitation of human motor activity, this is an
unacceptably high level. This fact makes it difficult to use high-torque, lowspeed motors with permanent magnets in gearless drives of exoskeletons.
It can be seen that the change in the inductance of the motors affects
only the proportional coefficient of PI - current regulators, so the essence
of the way to improve the characteristics of the engines is that we carry out
the correction of the proportional coefficients by the error of the motor
speed. Figure 5.9 shows the compensation scheme for inductance
changes.
In this case, you can choose a proportional coefficient of Eregulatory
as (5.2):
н =
тп,тп + ∆ (5.2)
Figure 5.9 – diagram of the compensation inductance
from formula (5.2) is defined as follows:
=
max
тп
−
,тп
(5.3)
Etalon
BDPT
Δ
K
p
=
K ω *
Δω
ПИ
ω
r
1
ω
r
2 ω ref
ω
r
2
1 / Kt
K
p
+
Δ
K
p
K i /
s
I
ref
I
abc
82
But this choice of proportional coefficient PI-controller is designed for
accurate knowledge of the law of change of inductance. In practice, this is
impossible. Therefore, we will build a control law in the adaptive control
class with a reference model (MRAC). The essence of the proposed
approach is to build a current regulator to compensate for the change in
inductance according to information about the instantaneous
electromotive force. In this case, the engine model will have the form of an
aperiodic link of the first order. The implementation scheme of the
algorithm is shown in figure 5.10, where W(s) – the transfer function of the
control object is indicated and is determined according to the formula (5.4);
Wm(s) – the reference transfer function of the control object and is
determined according to the formula (5.5):
() =
1
(5.4)
+
() =
1
+
(5.5)
Figure 5.10 – Scheme of mrac algorithm implementation
The adaptive algorithm has the form presented in the formula 5.6:
= − = − (5.6)
ПИ
I
ref I abc
+
- - +
u u
c
=
1
+
= 1
+
83
The output signal from the regulator is presented in the form of
formula 5.7, where
1 = 10 = ⁄ и 2 = 20 = − :
= 1. + 2. (5.7)
Figure 5.11 shows the General control scheme of a DC brushless
motor:
Figure 5.11 – General diagram of the control plug
Figure 5.12 presents the results of modeling the electric drive on the
BDP and the adaptive controller:
84
Figure 5.12 – simulation Results of the adaptive process
Figure 5.13 shows the simulation results when the mutual inductance
changes up to 20% depending on the angle of rotation of the rotor (K1=0.2):
Figure 5.13 – simulation Results at K1 = 0.2
85
Figure 5.14 shows the simulation results when the mutual inductance
changes up to 50% depending on the angle of rotation of the rotor (K1=0.5).
Figure 5.14 – simulation Results at K1 = 0.5
In both cases, the blue curves correspond to the drive variables in the
absence of dependence of mutual inductance on the angle of rotation of
the rotor.
The determining are based on the torques of the actuators of the links
of the exoskeleton from time to time (right lower graphs). As can be seen
from the waveforms, along with the ripple arising from the discreteness of
the master controller (blue curves) is quite a significant contribution to the
distortion of the management process introduces the factor of
dependence of vzaimoinduktivnosti of the rotation angle of the rotor –red
curves show the pulsation of the torque generated by both of the specified
prichinami. As mentioned above, this is an unacceptably high level of
pulsation moment for exoskeletons used in medical practice for the
rehabilitation of human motor activity.
86
5.2. Computer simulation of five-stage exoskeleton motion
Applied to modelling the movement of parts of the exoskeleton
interpolation with specified conditions. Least squares interpolation is
widely used to process experimental curves. Its meaning is as follows: let
the experimental table is given:
Table 5.2 – table with experimental data
Put it in accordance with the function of the form (5.8), where () basis functions; - coefficients to be determined:
(, 0, 1, … ) = 00() + 11() + + () (5.8)
In order to determine the coefficients we will look for such a
function
(, 0, 1, … ), deviation of the values from the given table of values
is minimal. The function (5.9) is constructed in the point least squares
method):
(0, 1, … , ) (, , , … ) − ]2 (5.9)
Эта функция геометрически представляет собой сумму квадратов
отклонений значений yi от значений аппроксимирующей функции
(, 0, 1, … ) в точках xi.
A necessary condition for the minimum of the function of many
variables is the equality of zero of its partial derivatives of the first order by
independent variables (5.10):
87
0
(, 0, 1, … ) − ]0() = 0
(
, 0, 1, … ) − ]1() = 0 (5.10)
…
{
(, 0, 1, … ) − ]() = 0
This system is a system of linear algebraic equations of order m+1 with
respect to unknowns (a0, ..., аm). Its solution takes the minimum of the
function (0, 1, … , ). We consider the numerical method and kinematic
connections.
Single-point phase
We will simulate the movement of one step of a five-step walking robot
in a single-step phase (i.e.
2,
= 0 ) by interpolation method with the given
necessary conditions: T – time of one step; Δx–step length; L0 – distance
from the hip joint to the OX axis t = 0; Δh(t) – the height of the end point A2
of the transferred leg above the stepping surface as a function of time;
initial and final States within one step
q(0) = -
q(T); 1 (2) , 1 (2) ~0.
For a given step length, the angle of deviation of the legs from the
vertical is calculated by the formula (5.11):
Δx
0= arctg(
2
0) (5.11)
And the speed of the exoskeleton according to the formula (5.12):
= Δx⁄T(m/s) (5.12)
Select the basis functions for interpolation in the form (5.13):
88
(, 0, 1) = 1 sin() + 2cos( ), где: =
2
⁄T (rad/s) (5.13)
During the movement of the robot, first perform the interpolation of
the generalized coordinates [α1 ,β1] of the supporting leg.
The generalized coordinates of the portable leg are from the geometry
of the mechanism.
Let at the moment t = ti the position of the support leg is determined
by the coordinates α1(ti),β1(ti)] and the height of the point A2 of the
transferred leg Δh(ti).The remaining coordinates of the portable leg are as
follows: two circles with
centers O2 , A2 and radii la ,lb respectively. It can be seen that the points
of intersection of these circles are possible positions of the Shin of the
transferred leg. We introduce the following notations: h1,h2 – distances
from the hip joint to points A1, A2 in figure 5.15:
Figure 5.15 – single-Point phase
Condition of existence of intersection points: h2≤ (la + lb.)
In the case of maintaining the equality h1 = h2= (la + lb) in a single step,
we obtain that α1(t)= α2(t) = β1(t) = β2(t), and we obtain symmetric solutions.
In the case of h2< (la + lb) in the process of movement, will be a pair of roots.
Y
X
O
R
1
A
1
A
2
, t =
ti
X
B
2
, t =
0 B 1
Δ
h ( t
i
)
B 2 t = 0 t =
t i t
= T
h
1 h 2
89
The process of movement certainly meets the conditions of change
Cartesian
coordinates
2
(
−1
) <
2
(), therefore, only one root is taken into
account.
In the case of h2 > (la + lb) there are no roots, and to obtain a solution,
we will increase the height of point А2 by increment Δh0, then re-search for
the intersection points until the condition of the roots existence is satisfied.
From the set of roots of the intersection point B2, we obtain a table of
data on the motion of the knee joint, from which the interpolation by the
method of least squares we obtain the coordinates of the knee joint in the
function of time.
• Two-point phase
Will accept a number of conditions: the ratio between the two phases
4:1; rear leg makes the repulsion of the body forward and the front leg
applies the brakes; the Range of changes of angles in joints of the back leg
is in the knee joint: ∆ = 12 - 150 , the anchor point: ∆ = 8 – 12 0; the
Maximum longitudinal
the component of the main reaction vector of the support can reach
0.25 P.
To interpolate the generalized coordinates of the back foot, we take
the basis functions 5.14, where =
2
⁄0.2T (rad/s):
(, 0, 1) = 1 sin() + 2cos () (5.14)
At the interval ∆ = 0.2 from the final state [α1(T), β1(T)] of the singlesupport phase with the specified conditions, the coordinates of the hind
leg as a function of time are obtained.
The motion of the foot suspension point is described by the following
equations (5.15,
5.16), where: 1, 2-constants, 1, 2 = 0.:
() =
1
+ sin1 + 1 (5.15)
() =
1
+ cos1 + 1 (5.16)
90
The time functions of the front leg [α2(t), β2(t)] are defined similarly and
are shown in figure 5.16:
Figure 5.16 – two-Support phase
For the results of modeling the dynamics of two-legged walk, we take
the parameters of exoskeletons from [7]:
Table 5.3 – Parameters of the exoskeleton
m(кг)
r(м)
l(м)
J
(кг.м)
Housing
48.6
0.386
0.6
11.3
Hips'
8.6
0.18
0.41
1.02
Shins
5.6
0.32
0.49
0.53
T = 0.5c; ∆ = 0.43(м).
1. Single-point phase
Figures 5.17 show the change in the coordinates of the links in time,
the angles between the vertical and the hip, the vertical and the Shin, as
well as the dependence of the reactions of the supports and internal forces
on time in the single-support phase:
Y
N
A 1 X
B
1
t
= 1 . 2 T
t = T
A
2 B 2 R 1
R
2
π
/
2
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